Computing the Braid Monodromy of Completely Reducible $n$-gonal Curves

TL;DR

This paper introduces a new rectangular braid diagram method for computing the braid monodromy of completely reducible n-gonal curves and uses it to calculate Alexander polynomials of their complements.

Contribution

It presents a novel RBD method for braid monodromy computation and an algorithm for Alexander polynomial calculation for n-gonal curves.

Findings

Successfully computes braid monodromy for specific n-gonal curves.

Provides an algorithm to determine Alexander polynomials using Burau representations.

Includes examples demonstrating the effectiveness of the methods.

Abstract

Braid monodromy is an important tool for computing invariants of curves and surfaces. In this paper, the \emph{rectangular braid diagram (RBD)} method is proposed to compute the braid monodromy of a completely reducible $n$-gonal curve, i.e. the curves in the form $(y-y_1(x))...(y-y_n(x))=0$ where $n\in \mathbb{Z}^{+}$ and $y_i\in \mathbb{C}[x]$. Also, an algorithm is presented to compute the Alexander polynomial of these curve complements using Burau representations of braid groups. Examples for each computation are provided.